CHAPTER XII. CRITERIA FOR DIVISIBILITY BY A GIVEN NUMBER. In the Talmud1,100a+& is stated to be divisible by 7 if 2a+b is divisible by 7. Hippolytosla, in the third century, examined the remainder on the division of certain sums of digits by 7 or 9, but made no application to checking numerical computation. Avicenna or Ibn Sina (980-1037) is said to have been the discoverer of the familiar rule for casting out of nines (cf. Fontes39); but it seems to have been of Indian origin.16 Alkarkhi1' (about 1015) tested by 9 and 11. Ibn Mus& Alchwarizmild (first quarter of the ninth century) tested by 9. Leonardo Pisanole gave in his Liber Abbaci, 1202, a proof of the test for 9, and indicated tests for 7, 11. Ibn Albanna17 (born about 1252), an Arab, gave tests for 7, 8, 9. In the fifteenth century, the Arab Sibt el-Maridinilc tested addition by casting out multiples of 7 or 8. Nicolas Chuquet10 in 1484 checked the four operations by casting out 9's. J. Widmann1/l tested by 7 and 9. Luca Paciuolo2 tested by 7, as well as by 9, the fundamental operations, but gave no rule to calculate rapidly the remainder on division by 7. Petrus Apianus2a tested by 6, 7, 8, 9. Robert Recorde26 tested by 9. Pierre Forcadel3 noted that to test by 7 = 10—3 we multiply the first digit by 3, subtract multiples of 7, add the residue to the next digit, then multiply the sum by 3, etc. Blaise Pascal4 stated and proved a criterion for the divisibility of any number N by any number A. Let r1? r2, r3,..., be the remainders obtained when 10, 10r1? 10r2,... are divided by A. Then JV = a+106+100c+ ... is divisible by A if and only if a-{-rib+r2c+... is divisible by A. Babylonian Talmud, Wilna edition by Romm, Book Aboda Sara, p. 96. 10M. Cantor, Geschichte der Math., ed. 3, I, 1907, 461. i*>Ibid., 511, 611, 756-7, 763-6. I'Cf. Carra de Vaux, Bibliotheca Math., (2), 13, 1899, 33-4. ldM. Cantor, Geschichte der Math., ed. 3, I, 1907, 717. "Scritti, 1, 1857, 8, 20, 39, 45; Cantor, Geschichte, 2, 1892, 8-10. x/Le Talkhys d'Ibn Albannd publiS et traduit par A. Marre, Atti Accad. Pont. Nuovi Lincei, 17, 1863-4, 297. Cf. M. Cantor, Geschichte Math., I, ed. 2, 757, 759; ed. 3, 805-8. ^Le Triparty en la science de nombres, Bull. Bibl. St. Sc. Math., 13, 1880, 602-3. lABehede vnd hubsche Rechnung.'.., Leipzig, 1489. 2Summa de arithmetica geomctria proportion! et proportionalita, Venice, 1494, f. 22, r. 2aEin newe. . .Kauffmans Rechnung, Ingolstadt, 1527, etc. 36The Grovnd of Artes, London, c. 1542, etc. 'L'Arithmeticqve de P. Forcadel de Beziers, Paris, 1556, 59-60. 4De numeris multiplicibus, presented to the Acade"mie Parisienne, in 1654, first published in 1665; Oeuvres de Pascal, 3, Paris, 1908, 311-339; 5, 1779, 123-134. 337